plane kinematics of rigid bodies - indian institute of ... notes/me101-lecture31-kd.pdf · plane...

Plane Kinematics of Rigid BodiesRigid Body• A system of particles for which the distances between the particles remain

unchanged.

• This is an ideal case. There is always some deformation in materials under the

action of loads. This deformation can be neglected if the changes in the shape

are small compared to the movement of the body as a whole.

Particle Kinematics • Developed the relationships governing the disp, vel, and accln of points as they

move along straight or curved paths.

Rigid Body Kinematics• Same relationships will be used. In addition, rotational motion of rigid bodies will

also be accounted for.

• Involves both linear and angular disp, vel, and accln.

• Discussion will be restricted to motion in a Single Plane (Plane Motion).

When all parts of the body move in parallel planes

Plane containing mass center is generally considered as plane of motion

and the body is treated as a thin slab whose motion is confined to the

plane of the slab.1ME101 - Division III Kaustubh Dasgupta

Plane Kinematics of Rigid Bodies

Plane MotionTranslationNo rotation of any line in body.

Motion of the body specified by

motion of any point in the body

≈ Motion of a single particle.

Rotation @ a Fixed AxisAll particles move in circular

paths @ axis of rotn. All lines

perpendicular to the axis of rotn

rotate through the same angle.

General Planar Motion

Combination of translation

and rotation

The actual paths of all

particles in the body are

projected on to a single

plane of motion.

2ME101 - Division III Kaustubh Dasgupta


Rotation• Described by angular motion

Consider plane motion of a rotating rigid body

since β is invariant

Therefore,

And, during a finite interval:

All lines on a rigid body in its plane of motion have the same angular

displacement, same angular velocity (ω), and same angular acceleration (α).

For plane Rotation: For rotation with constant

angular acceleration,

integrals of these eqns give:

+ve dirn for ω and α is the same as that chosen for θ

Particle Kinematics Rigid Body Kinematics

(s, v, a) (θ, ω, α)

At t = 0, θ = θ0, ω = ω0



Rotation @ a Fixed Axis•All points (except those on the axis) move

in concentric circles @ the fixed axis normal

to the plane of the figure

Vel of A using n-t for circular motion:

For particle circular motion For body rotational motion

rrvvrv tn ,0



• Rotation @ a Fixed Axis


Using the cross-product relationship of

vector notation:

The vector equivalent eqns:


Absolute Motion Analysis

•First approach to rigid body kinematics

•Similar to the absolute motion analysis of the constrained motion of

connected particles discussed in kinematics of particles (for pulley

configurations considered, the relevant velocities and accelerations were

obtained by successive differentiation of the lengths of the connecting

cables).

•In case of absolute motion analysis of rigid bodies, both linear and

angular velocities and linear and angular accelerations are required to be

considered.



Relative Motion Analysis: Velocity

•Second approach to rigid body kinematics

•In case of kinematics of particles, motion relative to translating axes:

Relative velocity due to Rotation: AB A’B’ during Δt ΔrB/A = - ΔrA/B


- First translation,

then rotation of the

body

x’-y’ :: non-rotating

reference axes


• Relative Motion Analysis: Velocity


To the non-rotating observer attached to B,

:: the body appears to undergo fixed axis rotation @ B

:: Point A executes circular motion

:: Equations developed for circular motions can be used


Relative Motion Analysis: VelocityRelative velocity due to Rotation

ΔrA/B = has the magnitude rΔθ as Δθ approaches zero

•ΔrA/B is accompanied by Δθ as seen from

translating axes x’-y’

Relative velocity Equation

Magnitude of the relative vel:

Using r to represent vector rA/B, relative velocity may be written as the vector:

From already derived vector eqn for velocity

ω is the angular velocity vector normal to the plane of the motion

Relative linear velocity is always perpendicular

to the line joining two points in question (A-B or

A’-B)9ME101 - Division III Kaustubh Dasgupta


Relative Motion Analysis: VelocityRelative velocity due to Rotation

• Visualizing the separate translation and rotation components

of the relative velocity equation

• Choosing B as the reference point

• Relative linear velocity is always perpendicular to the line joining two points in

question (A-B)

• ω is the absolute angular velocity of AB.



Relative Motion Analysis: VelocityExample

Solution: Choosing center O as the reference

point for relative velocity eqn since its

motion is given.

The relative velocity of A wrt O is observed

from the translating axes x-y attached to O.



Relative Motion Analysis: VelocityExample

Solution:

Direction of relative velocity is perpendicular to OA.

The angular velocity of OA is the same as that of the

wheel:

The vector sum vA can be calculated from law of

cosines:

Dirn of vA can be easily found out.

Point C momentarily has zero velocity and may also

be chosen as a reference point:

The problem may also be solved using vector formulation.

AC = 200sin30 + [(200cos30)2+(300)2]

= 436 mm



Instantaneous Center of Zero Velocity

•Relative Motion Analysis: velocity of a point on a rigid body in plane

motion = relative velocity due to rotation @ a convenient reference point

+ velocity of the reference point.

•The problem can also be solved by choosing a unique reference point

that momentarily has zero velocity.

This point lies on Instantaneous Axis of Zero Velocity (or

instantaneous axis of rotation), and intersection of this axis with the

plane of motion is known as Instantaneous Center of Zero

Velocity.

As far as velocities are concerned, the body may be considered to be

in pure rotation in a circular path @ the Instantaneous Axis of Zero

Velocity.

Locating the instantaneous center of zero velocity is important to

simplify the solution of many problems involving rigid body rotations



Instantaneous Center of Zero VelocityLocating the Instantaneous Center

• Assume that the dirns of absolute vel of any points A and B

on rigid body are known and are not parallel.

• If C is a point @ which A has absolute circular motion at the

instant considered, C must lie on the normal to vA through A.

• Similarly, point B can also have absolute circular motion @

C at the instant considered.

Intersection of two perpendiculars will give the absolute center of rotation at the

instant considered.

Point C is the instantaneous center of zero velocity and may lie on or off the body

Instantaneous center of zero velocity need not be a fixed point in the body or a

fixed point in the plane.

If vel of two points are parallel and the line joining the

points is perpendicular to the dirn of vel, the

instantaneous center is located by direct proportion.

As the parallel velocities become equal in magnitude, the IC

moves farther away from the body and approaches infinity in the

limit body stops rotating and translates only.



Instantaneous Center of Zero VelocityLocating the Instantaneous Center

If magnitude of velocity at one of the points on the rigid body

under general plane motion is known (vA)

Angular velocity of the body ω and linear velocity of every

point in the body can be easily obtained using:

ω = vA / rA this is also the angular velocity of every line in the body

velocity of B is vB = rBω = (rB / rA) vA

• Once the IC is located, the dirn of the

instantaneous velocity of every point in the

body is readily found since it must be

perpendicular to the radial line joining the

point in question with IC.

Velocity of all such points: v = rω

Note that the spokes are somewhat visible near IC,

whereas at the top of the wheel they become blurred.

Also note how the direction of velocity varies at different

point on the rim of the wheel.



Relative Motion Analysis: Velocity

Instantaneous Center of Zero Velocity

Motion of the Instantaneous Center

• IC changes its position in space and on the body as body changes its position

• Locus of the ICs in space is known as space centrode

• Locus of the ICs on the body is known as body centrode

• At the instant considered, the two curves are tangent at C

• The body centrode rolls on the space centrode during

the motion of the body.

• Although the IC of zero velocity is momentarily at rest,

its accln is generally not zero.

IC cannot be used as an IC of zero acceleration.

• Instantaneous center of zero acceleration exist for bodies in general plane

motion will not be discussed in ME101



Relative Motion Analysis: VelocityInstantaneous Center of Zero VelocityExample


Plane Kinematics of Rigid BodiesInstantaneous Center of Zero VelocityExample

Solution:

Directions of vel of A and B are tangent to their

circular paths @ fixed centers O and O’

• Intersection of the two perpendiculars to the

vel from A and B IC C for the link AB

AC = ABtan45 = 350tan45 = 350 mm

CD2 = 3502 + 1502 CD = 381 mm

CB2 = 3502 + 3502 CB = 350√2 mm

Angular vel of OB: ωOB = 10 rad/s

vB = r ω = 150√2x10 = 2121.3 mm/s

Ang vel of BC: ωBC = 2121.3/350√2 = 4.29 rad/s

BC is a line considered on the extended rigid

body ωBC = ωAC = ωDC = ωAB (clockwise)

vA = r ω = 350x4.29 = 1.5 m/s

vD = r ω = 381x4.29 = 1.632 m/s

Directions of velocities are shown.18ME101 - Division III Kaustubh Dasgupta

Plane Kinematics of Rigid BodiesInstantaneous Center of Zero Velocity

Example

Show the instantaneous center of

zero velocity for member BC.

Solution:

Point B follow circular path of motion

because link AB is subjected to

rotation about a fixed axis.

vB is perpendicular to AB

Motion of B causes the piston to move

horizontally.

vC is horizontal

Draw lines perpendicular to vB and vC

These lines intersect at IC.



Example

Show the instantaneous center of

zero velocity for the link CB.

Solution:

Points B and C follow circular path of motion

because links AB and DC are subjected to rotation

about a fixed axis.

vB and vC are perpendicular to AB and DC at the

instant considered

Radial lines drawn perpendicular to vB and vC are

parallel and Intersect at infinity

ωCB = vC /∞ = o

Link CB momentarily translates

An instant later, CB will move to a tilted position,

causing IC to move to some finite location.



Example

The cylinder rolls without slipping

betn the two moving plates.

Determine the location of IC and

calculate the angular velocity of

the cylinder and velocity of its

center C.

Solution:

Since no slipping occurs, velocities of points A and

B on cylinder will be same as vE and vD resp.

Since vA and vB are parallel, IC can be located by

proportionality of right triangles on line joining AB.

Direction of velocity of C and hence that of angular

velocity of cylinder can also be known from the

similar triangles.

.21ME101 - Division III Kaustubh Dasgupta


Example: Solution:

Dividing one eqn by other to eliminate ω:



Relative Motion Analysis: AccelerationEqn that describes the relative velocities of two points A and B in plane motion in

terms of nonrotating reference axes:

Differentiating wrt time:

Acceleration of point A is equal to vector sum of acceleration of point B and the

acceleration which A appears to have to a nonrotating observer moving with B.

Relative Acceleration due to Rotation

Consider two points on a rigid body. Since distance between them remains the

same, the observer moving with B perceives A to have circular motion about B.

• Since the relative motion is circular, relative acceleration term will have both

normal as well as tangential components.

• Normal component of accln will be directed from A

towards B due to change in direction of vA/B.

• Tangential component of accln will be perpendicular

to AB due to the change in the magnitude of vA/B.

aA and aB are the absolute accelerations of A and B.

Not tangent to the path of motion when the motion is curvilinear.



Relative Motion Analysis: AccelerationWe can write:

The magnitudes of the relative accln components:

Acceleration components in vector notations:

ω is the angular velocity of the body

α is the angular acceleration of the body

r is the vector locating A from B

The relative accln terms depend on the respective

absolute angular vel and absolute angular accln.

Alternatively: aB = aA + aB/A


plane kinematics of rigid bodies - indian institute of ... notes/me101-lecture31-kd.pdf · plane...

Documents